Problem with superscripts in limits

Question

Problem

My problem concerns the position of the variable n, specifically when used with limits as shown in the following output. The fragment x \in \mathbb{C}^{n} works as desired. but when this fragment is used in \argmin_{x \in \mathbb{C}^{n}}, the variable n has dropped. A red line is added to accentuate the drop of n.

Question

How should the second instance be written so that n has the same relative location in both instances?

Example

\documentclass[]{article}
\usepackage{amsmath,amssymb}
\DeclareMathOperator*{\argmin}{arg\,min}
\begin{document}
\begin{equation}
    x \in \mathbb{C}^{n}, \quad \argmin_{x \in \mathbb{C}^{n}}
\end{equation}
\end{document}

Output

Answer 1

You can force "uncramped" style by placing \mathbb{C}^{n} in scriptstyle (even though it already is).

\documentclass[]{article}
\usepackage{amsmath,amssymb}
\DeclareMathOperator*{\argmin}{arg\,min}
\begin{document}
\begin{equation}
    x \in \mathbb{C}^{n}, \quad \argmin\limits_{x \in {\scriptstyle\mathbb{C}^{n}}}%
\end{equation}
\end{document}